Recording Input Uncertainty and Applicability Limits

Why an Input Needs More Than a Point Value

A PBPK parameter table may contain a value such as \(f_u=0.034\), but that number alone does not say how precisely it was measured, how much observations varied, or whether the assay population resembles the population of interest. Recording those distinctions matters: uncertainty can affect how an input is used in analysis, while an applicability limit can make a precise measurement a poor choice for a particular modeling context.

The earlier material on parameter records and measurement provenance provides the setting: an input should remain traceable to its source and the measurement from which it was obtained. This tutorial focuses on documenting uncertainty and scope alongside that input, without turning every caveat into a numerical distribution.

Definition: Measurement uncertainty describes imprecision associated with a measured value or its estimate, such as sampling error in a mean or variation across assay replicates. Applicability limits describe reasons a measured value may not represent the target PBPK context, such as differences in population, species, disease state, assay system, or experimental conditions. These are related but distinct: a value can be estimated precisely within an assay and still have limited applicability elsewhere.

A useful input record therefore distinguishes the reported point estimate from the uncertainty summary and from the limits on interpretation. It should preserve what the source actually reports, identify the observations or experimental units behind a calculated summary, state units, and avoid implying that a reported range or standard error answers a question it was not designed to answer.

What to Record and What Each Summary Means

A standard error (SE) describes the estimated standard deviation of a statistic's sampling distribution under a specified sampling model. For a sample mean based on \(n\) independent observations with sample standard deviation \(s\), the estimated standard error is \(s/\sqrt{n}\). It describes uncertainty in the estimated mean under the stated assumptions; it is not the standard deviation of individual observations.

SE(\bar{x})=\frac{s}{\sqrt{n}}

A confidence interval adds a confidence procedure to the estimate. For independent observations from a normally distributed population with unknown standard deviation, a two-sided \(t\) confidence interval for the population mean is \(\bar{x}\pm t^*s/\sqrt{n}\), where \(t^*\) is the appropriate critical value with \(df=n-1\). The procedure has its stated long-run coverage when its assumptions and sampling design are appropriate. It is not a probability statement that the fixed population mean lies in this particular calculated interval.

An observed range is simply the smallest and largest values among the observations reported. It conveys the spread of those observations, but by itself does not provide a confidence level, standard error, or a distribution for future values. A range might instead be a source-reported plausible interval, a tested assay range, or a modeler's chosen scenario range; the record must label which meaning applies.

Recording principle: Preserve the source's uncertainty summary and its meaning. For any calculated summary, record the data basis, calculation, units, and assumptions. Separately state qualitative applicability limits. Do not calculate an uncertainty measure from insufficient information or label an observed range as a confidence interval.

For a practical PBPK parameter record, uncertainty documentation can include: the point estimate and units; the reported or calculated SE, confidence interval, or range; sample size and what counted as an independent observation; the method and assumptions; and a concise applicability note. If uncertainty is not reported, say so rather than filling the gap with an unsupported value.

Worked Examples

Worked Example: Confidence Interval for a Donor Mean

Suppose a hypothetical assay reports unbound fraction measurements from 12 independent healthy-donor samples. The mean is \(\bar{x}=0.0340\), and the sample standard deviation is \(s=0.0100\). Assume the donor samples were obtained using a design suitable for estimating the mean in the stated healthy-donor population, and that the measurements are independent and reasonably consistent with a normal sampling model. Calculate and document a 95% \(t\) interval.

The estimated standard error is:

SE(\bar{x})=\frac{0.0100}{\sqrt{12}}=0.00288675\approx 0.002887

There are \(df=12-1=11\) degrees of freedom. Using a 95% two-sided critical value of approximately \(t^*=2.201\), the margin of error, using the displayed rounded SE, is:

2.201(0.002887)=0.006354287\approx 0.006354

Thus the approximate interval is:

0.0340\pm 0.006354=(0.027646,\ 0.040354)\approx(0.02765,\ 0.04035)

The record should state that the mean unbound fraction was 0.0340, with \(SE=0.002887\) and an approximate 95% \(t\) confidence interval of 0.02765 to 0.04035, based on 12 independent healthy-donor samples. The interval quantifies uncertainty in the mean for the specified sampling context, under the stated assumptions. It does not establish that the same mean applies to patients with a different disease state, or that individual patients' values fall within the interval. Those are applicability or individual-variability questions requiring separate evidence.

Worked Example: Small Assay Series and a Wide Interval

In a hypothetical in-vitro assay, three independent runs give intrinsic-clearance estimates of 8.0, 9.0, and 10.0 \(\mu\text{L}/\text{min}/\text{mg}\). Treat the runs as independent observations from a population of comparable assay runs and use a \(t\) interval for their mean. First, the mean is:

\bar{x}=\frac{8.0+9.0+10.0}{3}=9.0\ \mu\text{L}/\text{min}/\text{mg}

The deviations from the mean are \(-1.0,\ 0,\) and \(1.0\). Their squared deviations sum to \(2.0\), so:

s=\sqrt{\frac{2.0}{3-1}}=1.0\ \mu\text{L}/\text{min}/\text{mg},\qquad SE(\bar{x})=\frac{1.0}{\sqrt{3}}=0.5774\ \mu\text{L}/\text{min}/\text{mg}

With \(df=2\), the 95% two-sided critical value is approximately 4.303. The margin, using the unrounded \(SE=1/\sqrt{3}\ \mu\text{L}/\text{min}/\text{mg}\), is \(4.303(1/\sqrt{3})=2.484338\approx2.484\ \mu\text{L}/\text{min}/\text{mg}\), giving:

9.0\pm2.484=(6.516,\ 11.484)\ \mu\text{L}/\text{min}/\text{mg}

The interval is wide relative to the estimate, in part because the sample is very small. It is a confidence interval for the mean under the stated model, not a range guaranteed to contain future run values. If the three runs were technical repeats of one preparation rather than independent experimental units, treating them as \(n=3\) independent observations would overstate the information about between-preparation uncertainty. The record must identify the experimental unit and should not silently treat technical replication as biological replication.

Worked Example: Recording an Observed Range Without Overinterpreting It

A hypothetical report gives five replicate assay estimates of a parameter, in arbitrary units: 0.021, 0.025, 0.028, 0.031, and 0.035. Their mean is \(0.140/5=0.028\), and the observed range is 0.021 to 0.035. To illustrate the distinction, the sample standard deviation is:

s=\sqrt{\frac{(0.021-0.028)^2+(0.025-0.028)^2+(0.028-0.028)^2+(0.031-0.028)^2+(0.035-0.028)^2}{5-1}} =\sqrt{\frac{0.000116}{4}} =0.005385

The range is a summary of these five observed measurements; it is not a 95% confidence interval and does not, by itself, specify a distribution for uncertainty sampling. The calculated \(s\) describes their observed dispersion, while an SE for their mean would be \(0.005385/\sqrt{5}=0.002408\), under an independent-replicate interpretation. A parameter record should preserve the range as reported and identify the replicate basis. If the source used a different meaning of “range,” such as a tested concentration interval, record that definition instead.

Suppose the assay was performed in a specific in-vitro system. The record should separately note that this observed replicate range describes that assay context and does not establish a human population range. The limitation is qualitative unless suitable evidence supports a numerical adjustment or distribution.

A Short Justification for the Standard Error

The standard error formula follows from the variance of a mean. Let \(X_1,\ldots,X_n\) be independent observations with common variance \(\sigma^2\), and let \(\bar{X}=(X_1+\cdots+X_n)/n\). Independence makes the covariance terms zero, so:

\operatorname{Var}(\bar{X}) =\operatorname{Var}\left(\frac{1}{n}\sum_{i=1}^{n}X_i\right) =\frac{1}{n^2}\sum_{i=1}^{n}\operatorname{Var}(X_i) =\frac{n\sigma^2}{n^2} =\frac{\sigma^2}{n}

Therefore the standard deviation of the sampling distribution of \(\bar{X}\) is \(\sigma/\sqrt{n}\). Since \(\sigma\) is generally unknown, \(s/\sqrt{n}\) estimates it. This derivation depends on independence and a common variance; repeated measurements, clustered samples, or unequal sampling designs may require a different analysis. A record should not present \(s/\sqrt{n}\) as justified merely because a source reports a standard deviation and a count.

Applicability Limits, Diagnostics, and Alternatives

Applicability notes answer a different question from an interval: what evidence supports using this measured value for the intended model context? Relevant qualifiers may include species, age, sex, disease status, tissue source, assay system, temperature, pH, substrate concentration, or whether the source measured a population mean or a single preparation. These notes should be specific enough to guide review. “Limited applicability” is less useful than “measured in pooled healthy-donor microsomes; evidence in the target disease population was not reported.”

Check for warning signs before transferring a number into a PBPK input table:

The reported SE is mistaken for the spread of individual measurements, or the reported standard deviation is entered as the SE.
The stated \(n\) counts technical repeats, while the intended inference concerns independent donors, animals, or preparations.
A confidence interval is copied without its confidence level, method, population, or units.
A range is treated as a probability distribution or confidence interval even though its construction is unknown.
The estimate is reported with many digits despite substantial uncertainty or limited assay precision.
The point estimate is precisely measured, but population or assay differences are omitted from the applicability note.

When choosing how to summarize evidence, keep each option's role clear:

Summary	What it conveys	What it does not establish by itself
Standard error	Estimated sampling uncertainty of a statistic, given a sampling model	Individual variability or transferability to a different population
Confidence interval	Uncertainty about a parameter using a specified interval procedure	A probability that the fixed parameter lies in the particular interval
Observed range	Smallest-to-largest values in a specified set of observations	A confidence level or a distribution for future values
Qualitative limitation	A documented boundary on interpretation or use	A quantitative uncertainty distribution unless supported by data

A practical alternative to forcing a numerical uncertainty summary is to report the point estimate with “uncertainty not reported” and describe the scope limitation. Conversely, where data and assumptions support an SE or confidence interval, recording only a vague label such as “variable” discards useful evidence. The appropriate summary is the one that matches the design and the question, not the one that looks most quantitative.

Common Mistake and Key Takeaway

Wrong reasoning: “The assay range is 0.021 to 0.035, so the parameter has a 95% uncertainty interval of 0.021 to 0.035 and applies to the target population.” This confuses an observed range with a confidence interval and assumes transferability without evidence.

Correction: State that 0.021 to 0.035 is the observed range among the specified assay replicates. Give a confidence interval only if a suitable procedure and assumptions justify one, and document target-population or assay applicability separately. A number's precision in its original measurement context does not remove the need to assess its scope.

Key takeaway: Record the value, uncertainty summary, data basis, units, assumptions, and applicability limits as distinct pieces of information. Quantify uncertainty only when the design supports the calculation; otherwise preserve the limitation plainly and do not invent precision.

Check Your Understanding

Answer each question, then compare with the brief solutions.

A mean is based on 25 independent observations with \(s=0.20\) units. What is its estimated SE?
In one sentence, distinguish a 95% confidence interval for a mean from the observed range of the sample.
Why might five technical repeats from one donor not justify an SE for a population of donors?
A source reports a precise assay estimate but gives no uncertainty summary and studied a different population from the target. What should the input record say?

Solutions: (1) \(SE=0.20/\sqrt{25}=0.040\) units. (2) A confidence interval is produced by a procedure intended to cover the population mean at the stated long-run rate under its assumptions; the observed range only gives the minimum and maximum in the sample. (3) Technical repeats may not be independent donor-level observations, so using them as the population sample size can misrepresent the sampling design. (4) Preserve the reported estimate and units, state that uncertainty was not reported, and explicitly note the population mismatch as an applicability limit rather than fabricating a numerical adjustment.